Axial ratio is the difference between the 'horizontal' and 'vertical' (or otherwise orthogonal) components. The most appropriate 'units' would be dB, but a straight factor could be used as well (it is just a ratio, a unitless number).
Imagine that you create the circular polarization using two properly phased and spaced orthogonal dipoles. Imagine that one dipole isn't working as well as the other - perhaps due to additional cable loss in the phasing network. The circle will not be 'round'. One axis will be 'bigger' than the other axis.
As far as calculating it, it will typically be based on measurements of each axis using a linear sense antenna. A linear antenna should detect equal signal at any orientation in the plane of the wave - if the incoming signal was perfectly circular.
Obviously the imperfect circularity could be introduced in the environment (after the signal leaves the source) - especially reflections.